Welfare-Maximizing Monetary Policy Under Parameter Uncertainty

Transcription

1 Welfare-Maximizing Monetary Policy Under Parameter Uncertainty Rochelle M. Edge, Thomas Laubach, and John C. Williams March 1, 27 Abstract This paper examines welfare-maximizing monetary policy in an estimated dynamic stochastic general equilibrium model of the U.S. economy where the policymaker faces uncertainty about the true values of model parameters. Uncertainty about parameters describing preferences and technology implies not only uncertainty about the dynamics of the economy. In addition, it implies uncertainty about the model s utility-based welfare criterion and model dynamics but also uncertainty about the natural rate of output that the central bank should aim to achieve absent nominal rigidities and the natural rate of interest that is consistent with this level of output equalling its natural rate absent nominal rigidities. We analyze the characteristics and performance of alternative monetary policy rules given the estimated uncertainty regarding parameter estimates. We find that the natural rates of interest and output are imprecisely estimated. We then show that optimal policies under parameter uncertainty respond primarily to indirect signals regarding natural rates extracted from observed prices and wages and the level of hours. JEL Code: E5 Keywords: technology shocks, monetary policy rules, natural rate of output, natural rate of interest. Board of Governors of the Federal Reserve System, rochelle.m.edge@frb.gov; Board of Governors of the Federal Reserve System, tlaubach@frb.gov; and Federal Reserve Bank of San Francisco, john.c.williams@sf.frb.org (corresponding author). We thank seminar participants at the European Central Bank and Humboldt Universität Berlin for comments on this research project and paper. The views expressed herein are those of the authors and do not necessarily reflect those of the Board of Governors of the Federal Reserve System, its staff, or the management of the Federal Reserve Bank of San Francisco.

2 1 Introduction This paper examines welfare-maximizing monetary policy in an estimated dynamic stochastic general equilibrium model of the U.S. economy where the central bank faces uncertainty about the true values of model parameters. In this framework, parameter uncertainty implies uncertainty not only about the dynamics of the economy, but also about the central bank loss function and the natural rates of interest and output. Household welfare is maximized when output equals its natural rate, i.e. the value that would obtain absent nominal rigidities, at which point the real interest rate equals its corresponding natural rate. These natural rates change over time in response to shocks and their dynamic behavior depends on the model parameters describing preferences and technology. Owing to the presence of sticky prices and wages, the first-best outcome is not attainable and the central bank faces a tradeoff between minimizing deviations of output from its natural rate, the output gap, and minimizing fluctuations in price and wage inflation, where the relative weights on the three objectives depend on the model parameters. Thus, the model parameters jointly determine the dynamics of the economy, the dynamics of the natural rates, and the weights in the welfare-maximizing central bank s objective function. We analyze the implications of parameter uncertainty on the design of implementable optimal monetary policies and, in particular, how it affects the usefulness of measures of the natural rates of output and interest in determining policy decisions. Giannoni (22), Levin and Williams (25), and Levin, Onatski, Williams and Williams (25; henceforth LOWW) have explored aspects of monetary policy under parameter uncertainty in microfounded models where the central bank aims to maximize household welfare, but these papers do not explicitly analyze the usefulness of natural rates as guides for policy. We use the estimated covariance of model parameters as a measure of parameter uncertainty. We first show that the natural rates of output and interest are imprecisely estimated owing to parameter uncertainty. We then show that under parameter uncertainty, optimal policies rely less on estimates of the output gap, and more on prices and wages than would be optimal if natural rates were known. We also find that policies that respond to the level of labor hours do as well or better than policies that respond to hours or output gaps. This paper contributes to the large literature that examines the usefulness of measures of the natural rates in the conduct of monetary policy (see Orphanides and Williams, 22, and citations therein). The past research has generally been conducted with an ad hoc 1

3 policy objective and has treated movements in natural rates as exogenous and not directly related to parameter uncertainty. 1 That literature has generally found that in the context of simple monetary policy rules, monetary policy should respond less to variables related to natural rates like the output gap and the natural rate of interest and more to indirect measures, such as the rate of inflation and the rate of output growth (see Orphanides and Williams (22) and references therein). In contrast to the previous literature, we consider general parameter uncertainty that includes the slopes of macroeconomic relationships that affect the dynamic responses of natural rates to shocks. Thus, natural rate uncertainty is intrinsically connected to parameter uncertainty and certainty equivalence does not apply. Despite the different analytical framework used in this paper, our main results are similar in spirit to those in the previous literature; natural rates are a potentially unreliable guide for policy and a more robust strategy is to respond to other, better measured, variables like the rates of wage and price inflation and the level of hours. The remainder of the paper is organized as follows. Section 2 describes the model. Section 3 describes the model estimation and reports the results. Section 4 examines optimal monetary policy assuming model parameters are known. Section 5 considers optimal policy under parameter uncertainty. Section 6 concludes. 2 The Model In this section, we develop a fairly standard closed-economy model in which households choose consumption and set wages for their differentiated types of labor services, and in which firms produce using a CES aggregate of households labor services as input and set prices for their differentiated products. The dynamics of nominal and real variables are determined by the resulting first-order conditions of optimizing agents. We allow for various frictions such as habit formation and adjustment costs that interfere with instantaneous full adjustment in response to shocks. We begin by presenting preferences and technology and 1 In such a setting, natural rate uncertainty is a form of additive uncertainty and, under certain stringent conditions, has no implications for the optimal monetary policy. In particular, if the data generating process for the natural rates is known, then the separation principle and certainty equivalence applies. In that case, the optimal estimates of the natural rates are inserted into the optimal policy rule and the parameters of the optimal policy are unaffected by natural rate uncertainty. 2

4 then describe firms and households optimization problems. We log-linearize the equations describing the dynamic behavior of the economy, as described in Appendix A. We denote the log of variables by lower case letters. 2.1 The production technology The economy s final good, Y f,t, is produced according to the Dixit-Stiglitz technology, ( 1 Y f,t = ) θp θp 1 θp 1 Y f,t (x) θp dx, (1) where the variable Y f,t (x) denotes the quantity of the xth differentiated goods used in production and θ p is the constant elasticity of substitution between the differentiated production inputs. Final goods producers obtain their differentiated production inputs used in production from the economy s differentiated intermediate goods producers who supply an output Y m,t (x). Not all of the differentiated output produced by the intermediate goods producers is realized as inputs into final goods production; some is absorbed in price formulation, following the adjustment cost model of Rotemberg (1982). between Y f,t (j) and Y m,t (j) is given by, Y f,t (j) = Y m,t (j) χ p 2 Specifically, the relationship ( ) Pt (j) 2 P t 1 (j) Π p, Y m,t. (2) The second term in (2) denotes the cost of setting prices. This is quadratic in the difference between the actual change in price and steady-state change in price, Π p,. Our choice of quadratic adjustment costs for modeling nominal rigidities contrasts with that of many other recent studies, which rely instead on staggered price-and wage-setting in the spirit of Calvo (1983) and Taylor (198). We prefer the quadratic adjustment cost approach over staggered price- and wage-setting because the latter imply heterogeneity among agents. Partly for this reason, models utilizing staggered price and wage setting typically assume that utility is separable between consumption and leisure, in which case perfect insurance among households against labor income risk eliminates heterogeneity of their spending decisions. By contrast, if wages are staggered and household utility is nonseparable, differences across households in labor supply (which will result due to differences in wages set) lead to differences across household in the marginal utility of consumption (and hence consumption), even if perfect insurance is able to equalize wealth across households. 3

5 The quadratic adjustment cost model allows us to avoid heterogeneity across agents. In any case, the resulting price and wage inflation equations are very similar to those derived from Calvo-based setups as in Erceg, Henderson, and Levin (2). The differentiated intermediate goods, Y m,t (j) for j [, 1], are produced by combining each variety of the economy s differentiated labor inputs that are supplied to market activities (that is, {L y,t (z)} for z [, 1]). The composite bundle of labor, denoted L y,t, that obtains from this aggregation implies, given the current level of technology A t, the output of the differentiated goods, Y m,t. Specifically, production is given by, ( 1 Y m,t (j) = A t L y,t (j) where L y,t (j) = ) θw 1 θ w θw 1 Θ L y,t (x, j) l,t dx. (3) where θ w is the constant elasticity of substitution between the differentiated labor inputs. The log-level of technology, A t, is modeled as a random walk: ln A t = ln A t 1 + ɛ t (4) where ɛ t is an i.i.d. innovation. We abstract from trend growth in productivity. Throughout this paper, we restrict our analysis to permanent shocks to the level of technology. 2.2 Preferences Households derive utility from their purchases of the consumption good C t and from their use of leisure time, equal to what remains of their time endowment L after L u,t (i) L hours of labor are supplied to non-gratifying activities. We assume the household members live forever and there is no population growth. Its preferences exhibit an endogenous additive habit (assumed to equal a fraction η [, 1] of its consumption last period) and are nonseparable between consumption and leisure. 2 Specifically, preferences of household i are given by E 1 1 σ [ β t (C t (i) ηc t 1 (i))( L L u,t (i)) ζ] 1 σ, (5) t= where β is the household s discount factor, and ζ is a measure of the utility of leisure. The economy s resource constraint implies that 1 C t(x)dx Y f,t, where Y f,t denotes the output of the economy s final good. 2 Basu and Kimball (22) argue that nonseparability between consumption and leisure has substantial empirical support. 4

6 Non-gratifying activities include supplying L y,t hours to the labor market and devoting time to setting wages. Consequently, we define L u,t (i) as L u,t (i) = L y,t (i) + χ w 2 ( ) Wt (i) 2 W t 1 (i) Π w, L u,t. (6) The second term in (6) denotes the cost of setting wages in terms of labor time and is analogous to the cost of setting prices. 2.3 Firms optimization problems The final goods producing firm, taking as given the prices set by each intermediate-good producer for their differentiated output, {P t (j)} 1 j=, chooses intermediate inputs, {Y f,t(j)} 1 j=, so as to minimize the cost of producing its final output Y f,t, subject its production technology, given by equation (1). Specifically, the competitive firm in each sector solves 1 ( 1 ) θp θp 1 θp 1 min P t (x)y f,t (x)dx s.t. Y f,t Y f,t (x) θp dx. (7) {Y f,t (j)} 1 j= This problem implies a demand function for each of the economy s intermediate goods given by Y f,t (j) = (P t (j)/p t ) θ p Y f,t, where the variable P t is the aggregate price level, defined by P t = ( 1 (P t(x)) 1 θ p dx) 1 1 θp. Each intermediate firm chooses the quantities of labor that it emply use for production and the price that it will set for its output. It is convenient to consider these two decisions as separate problems. In the first step of the problem firm j, taking as given the wages {W t (i)} 1 i= set by each household for its variety of labor, chooses {L y,t(i, j)} 1 i= to minimize the cost of attaining the aggregate labor bundle L y,t (j) that it will ultimately need for production. Specifically, the materials firm j solves: 1 min {L y,t (i,j)} 1 i= W t (x)l y,t (x, j)dx s.t. Y m,t (j) A t ( 1 ) θw L y,t (x, j) θ w 1 θw 1 θw dx (8) This cost-minimization problem implies that the economy-wide demand for type i labor is L y,t (i) = 1 L y,t(i, x)dx = (W t (i)/w t ) θ w (1/A t ) 1 Y m,t(x)dx where W t denotes the aggregate wage, defined by W t = ( 1 (W t(x)) 1 θ 1 w dx) 1 θw. The marginal cost function of producing the intermediate goods is MC t (j) = W t /A t. In setting its price, P t (j), the intermediate good producing firm takes into account the demand schedule for its output that it faces from the final goods sectors and the fact as summarized in equation (2) that by resetting its price it reduces the amount of its 5

7 output that it can sell to final goods producers. The intermediate-good producing firm j, taking as given the marginal cost MC t (j) for producing Y m,t (j), the aggregate price level P t, and aggregate final-goods demand Y f,t, chooses its price P t (j) to maximize the present discounted value of its profits subject to the cost of re-setting its price and the demand curve it faces for its differentiated output. Specifically, the firm solves, max {P t(j)} t= subject to E t= Y f,t (j)=y m,t (j) χ p 2 β tλ { } c,t (1 + ς θy,t )P t (j)y f,t (j) MC t (j)y m,t (j) P t ( ) Pt (j) 2 ( P t 1 (j) Π Pt (j) p, Y m,t and Y f,t(j)= P t ) θp Y f,t, In (9) the discount factor that is relevant for discounting nominal revenues and costs between periods t and t + j is E t β j Λ c,t+j /P t+j Λ c,t/p t, where Λ c,t is the household s marginal utility of consumption in period t. The parameter ς θ,y is a subsidy (equal to (θ p 1) 1 ), which ensures that in the absence of nominal rigidities the model s equilibrium outcome is Pareto optimal. (9) 2.4 Households optimization problem The household taking as given the expected path of the gross nominal interest rate R t, the price level P t, the aggregate wage rate W t, its profits income, and its initial bond stock B i,, chooses its consumption C t (i) and its wage W t (i) to maximize its utility subject to its budget constraint, the cost of re-setting its wage, and the demand curve it faces for its differentiated labor. Specifically, the household solves: 1 max E {C t (i),w t (i)} t= 1 σ subject to E t [ t= β Λ ] c,t+1/p c,t+1 B t+1 (i) Λ c,t /P c,t β t Ξ c,t [ (C t (i) ηc t 1 (i))( L L u,t (i)) ζ] 1 σ =B t (i) + (1 + ς θ,l )W t (i)l y,t (i) + Profits t (i) P t C t (i), L y,t (i) = L u,t (i) χ ( ) w Wt (i) 2 2 W t 1 (i) Π w, L u,t, and ( ) Wt (i) θw 1 L y,t (i)= L y,t (i, j)dj. (1) W t The parameter ς θ,l in the household s budget constraint is a subsidy (equal to (θ w 1) 1 ), which ensures that in the absence of nominal rigidities the model s equilibrium outcome is 6

8 Pareto optimal. The variable B t (i) in the budget constraint is the state-contingent value, in terms of the numeraire, of household i s asset holdings at the beginning of period t. We assume that there exists a risk-free one-period bond, which pays one unit of the numeraire in each state, and denote its yield that is, the gross nominal interest rate between periods ( t and t + 1 by R t E t β Λ ) c,t+1/p t+1 1. Λ c,t /P t Profits in the budget constraint are those rebated from firms, which are ultimately owned by households. 2.5 Natural rate variables Our model has a counterpart in which all nominal rigidities are absent, that is, prices and wages are fully flexible. In this model the cost minimization problems faced by the final goods producing firm and the intermediate goods producing firms continue to be given by equations (7) and (8). The intermediate goods producing firms profit maximization problem is similar to equation (9) but with the price adjustment cost parameter χ p set to zero. Likewise the households utility maximization problem is given by equation (1) but with the wage adjustment cost parameter χ w set to zero. We refer to the level of output and real one-period interest rate in this equilibrium as the natural rate of output, Ỹt, and interest, R t. We also define log deviations of these variables from their steady-state values, ỹ t log Ỹt log Y and r t log R t log R. These natural rates are functions of our model s structural shocks and are derived in Appendix A. 2.6 Monetary authority In the model with nominal rigidities we assume that the central bank uses the short-term interest rate as its instrument, as discussed in section Equilibrium Our complete model consists of the first-order conditions (derived in Appendix A) describing firms optimal choice of prices and households optimal choices of consumption and wages, the production technology (3), the monetary policy rule, the market clearing conditions Y t (j) = 1 C j,t(i)di j and L t (i) = 1 L i,t(j)dj i, and the law of motion for aggregate technology (4). We now turn to the parametrization of our model. 7

9 3 Estimation We estimate several of the structural parameters of our model using a minimum distance estimator. Specifically, we estimate a VAR on quarterly U.S. data using empirical counterparts to the theoretical variables in our model, and identify two of the model s structural shocks using identifying assumptions that are motivated by our theoretical model. We then choose model parameters so as to match the impulse responses to these two shocks implied by the model to those implied by the VAR. 3 Estimation of model parameters by impulse response function matching has sometimes been criticized for being ad hoc in selecting which properties of the data the model has to match. While we agree that it would also be interesting to explore full information estimation methods, we nevertheless think that the estimation undertaken here is valuable precisely because we focus on properties of the data that have a clear structural interpretation. A more serious issue is the potentially weak identification of several model parameters. Although Canova and Sala (26) explore this problem specifically in the context of IRF matching, other estimation strategies are similarly susceptible to this problem. In this section we first describe the VAR and the identification of the two shocks, and then discuss our parameter estimates. 3.1 VAR specification and identification The specification of our VAR is determined by the model developed in the previous section and our identification strategy for the structural shocks. Concerning the latter, we follow Galí (1999) and assume that the technology shock is the only shock that has a permanent effect on the level of output per hour. The monetary shock is identified by a standard restriction on contemporaneous responses. Our model and identifying assumptions combined suggest the inclusion of five variables in the VAR: the first difference of log output per hour, price inflation (the first difference of the log of the GDP deflator), the log labor share, the first difference of log hours per person, and the nominal funds rate. Output per hour, the labor share, and hours are the Bureau of Labor Statistics (BLS) measures for the nonfarm business sector, where the labor share is computed as output per hour times the deflator 3 Recent applications of this estimation strategy are Rotemberg and Woodford (1997), Amato and Laubach (23), and Christiano, Eichenbaum, and Evans (25). 8

10 for nonfarm business output divided by compensation per hour. 4 Population is the civilian population age 16 and over. Letting Y t denote the vector of variables in the VAR, we view the data in the VAR as corresponding, up to constants, to the model variables Y t = [ (y t l t ), π t, y t l t w t, l t, r t ] (11) where lower case letters denote logs of the model variables. We estimate the VAR over the sample 1966q2 to 26q2, including four lags of each variable. Details of the implementation of the identification scheme are provided in Appendix B. A potentially controversial aspect of our specification is the inclusion of hours per capita in first differences. Recent years have witnessed a vigorous debate among macroeconomists whether hours worked increase or decline following a technology shock. Francis and Ramey (25) and Altig et al. (22) have attributed differences in results among different studies to the issue whether hours per capita are included in levels in which case the level of hours is usually found to rise immediately following a technology shock or whether hours enter in first differences or some other detrended form in which case the level of hours is often found to decline during the first few quarters following the shock. We do not attempt to decide this issue in this paper, but accept the first difference specification for the present purposes. The dashed lines in the panels of Figure 1 show the impulse responses to a permanent one percent increase in the level of technology. The dashed-dotted lines present one-standard deviation bands around the impulse responses, computed by bootstrap methods. 5 Upon impact, output immediately rises about half-way to its new steady-state level, whereas hours worked decline by about 1/4 percent. Over the following eight quarters, output completes its adjustment while hours worked return to their original level. Interestingly, the response of inflation to a technology shock suggests only a limited role for price stickiness, with inflation declining upon impact by almost a percentage point. Wage rigidity, by contrast, seems to be more important, as the initial response of the real wage is driven by the initial 4 By contrast, Altig et al. (22) and Galí, López-Salido, and Vallés (23) compute labor productivity by dividing real GDP by total hours in the nonfarm business sector, which could be problematic because of the trending share of nonfarm business output in GDP. 5 To prevent the standard error bands from diverging over time, we discard draws for which the implied reduced-form VAR was estimated to be unstable, such as draws for which the largest eigenvalue of the coefficient matrix in the reduced form, written in companion form, exceeds.99. In total, about 14 percent of all draws are being rejected. 9

11 price response, not nominal wage adjustment. The real wage completes its adjustment over the following two quarters. The estimated response of monetary policy is to accommodate the increase in output by keeping the real funds rate on balance unchanged. Figure 2 shows the impulse responses of the variables to a one percentage point positive funds rate shock. The estimated responses of output, the real wage, and inflation to a funds rate shock are consistent with many studies on the effects of monetary policy. Output falls within three quarters by about 3/4 percent in response to one percentage point increase in the funds rate that takes eight quarters to die out. Hours decline closely in line with output, and the real wage falls. The response of inflation exhibits a price puzzle that lasts for two quarters; thereafter, inflation declines for six quarters, to about.3 percent below its original level. The responses to a funds rate shock are more precisely estimated than the responses to the technology shock. 3.2 Model parameter estimates With the VAR impulse responses to a funds rate shock and a technology shock in hand, we proceed to estimate the structural and monetary policy parameters of our model. First, we calibrate four model parameters that have little effect on the dynamic responses to shocks. We set the discount factor, β =.9924, which corresponds to discounting the future at a 3 percent annual rate. We normalize the time endowment to unity. We set the steady-state rates of price and wage inflation to zero. Finally, we set both aggregation parameters θ w and θ p to 6, following LOWW (25). The remaining parameters are estimated by minimizing the squared deviations of the responses of the five variables [y t, π t, w t, l t, r t ] implied by our model from their VAR counterparts. The IRFs of these five variables in quarters through 8 following a technology shock in quarter, and in quarters 1 through 8 following a funds rate shock (the response in the impact quarter being constrained by the identifying assumption) provide a total of 85 moments to match. These moments are weighted inversely proportional to the standard error around the VAR responses, as in Christiano et al. (25). This has the effect of placing more weight on matching the impulse responses to the monetary shock, which, as noted before, are estimated with greater precision than the impulse responses to the technology shock. For purposes of model estimation, we assume that monetary policy is set according to 1

12 Table 1: Parameter Estimates Model Point Standard Correlation with Parameter Estimate Error σ ζ η κ w κ p σ ζ η κ w κ p φ r.89.1 φ π a simply policy rule in which the interest rate depends on the lagged interest rate and the current inflation rate only: r t = φ r r t 1 + (1 φ r )φ π π p,t + ɛ r,t, where ɛ r,t is an i.i.d. monetary policy shock. 6 Note that we have suppressed the constant that incorporates the steady-state levels of the interest and inflation rate. In addition, because the parameters θ w and χ w appear only as a ratio in the linearized version of the model (see Appendix A), they are not separately identified; the same is the case for the parameters θ p and χ p. We therefore estimate the ratios κ w = (θ w 1)(1 + ς θ,w )/(χ w Π w, ) = θ w /(χ w Π w, ) and κ p = (θ p 1)(1 + ς θ,p )/(χ p Π p, ) = θ p /(χ p Π p, ). In the end, we estimated seven free parameters: {σ, ζ, η, κ w, κ p, φ r, φ π }. The estimated parameters and associated standard errors are shown in the first two columns of Table 1. The correlation coefficients of the structural parameter estimates are shown in the final five columns of the table. The covariance matrix of the estimates is computed using the Jacobian matrix from the numerical optimization routine and the empirical estimate of the covariance matrix of the impulse responses from the bootstrap. As discussed before, one feature of both sets of impulse responses is that real output and hours adjust gradually in response to the shocks. In the case of a permanent technology 6 Preliminary estimation results indicated a slightly negative, but near zero, response of monetary policy to the output gap, perhaps because the theoretical notion of the output gap in our model bears little resemblance to measures of the output gap used by policymakers. In the results reported in the paper, we constrained the response to the output gap to zero. 11

13 shock, Rotemberg and Woodford (1996) demonstrated that DSGE models without intrinsic inertia will not display such hump-shaped patterns; instead, these variables jump on impact and adjust monotonically to their new steady-state values. We therefore find a significant role for habit persistence. Our estimate of the habit parameter η is somewhat smaller than those estimated by Fuhrer (2), Smets and Wouters (23) and Christiano et al (25), but slightly larger than that estimated by LOWW (25). The estimate of σ is higher than typical estimates based on macroeconomic data, but this estimate is very imprecise. As noted before, the VAR responses of real wages and inflation differ substantially depending on the source of the shock: rapid responses to technology shocks, and sluggish ones to funds rate shocks. This is a feature that our price and wage specification cannot deliver. Our estimates of κ w and κ p imply that wages are very slow to adjust, but prices adjust relatively rapidly to fundamentals. The evidence for relatively flexible prices comes from the IRFs to the technology shock; indeed, the IRFs to monetary policy shocks alone suggest very gradual price adjustment, consistent with the findings of Christiano et al (25). Despite the greater weight placed on matching the more tightly estimated responses of inflation and real wage to the funds rate shock, our model does better at matching the responses to a technology shock, as shown by the solid lines in figures 1 and 2. Our estimates of the parameters of the monetary policy rule, φ r and φ π, are broadly consistent with the findings of many other studies that estimate monetary policy reaction functions, such as that of Clarida, Galí, and Gertler (2). The preference parameters, especially σ and ζ, are relatively imprecisely estimated, while those associated with wage and price adjustment costs are estimated with a great deal of precision. The structural parameter estimates are highly correlated with one another, as seen in the table. The estimates of σ and ζ are highly negatively correlated, which is not surprising given that these two parameters enter multiplicatively in the utility of leisure. More interestingly, the estimate of σ is negatively correlated with the estimated values of κ p and κ w, which will have implications for the welfare costs under parameter uncertainty, as discussed below. 12

14 4 Welfare and Optimal Monetary Policy In this section we compute the optimal policy response to a technology shock assuming all model parameters are known. We assume that the central bank objective is to maximize the unconditional expectation of the welfare of the representative household. We further assume that the central bank has the ability to commit to future policy actions; that is, we examine optimal policy under commitment, as opposed to discretion. We consider only policies that yield a unique rational expectations equilibrium. By focussing only on technology shocks, we are arguably examining only a relatively small source of aggregate fluctuations in output and wage and price inflation and hence welfare losses. For example, LOWW (25), using a medium-scale DSGE model, find that other shocks, especially those to price and wage markups, have much larger effects on welfare than technology shocks. In order to conduct welfare-based monetary policy analysis incorporating other sources of fluctuations, we would need to take a stand on the precise source and nature (i.e., distortionary vs. fundamental) of the other shocks to the economy, as discussed in LOWW (25). This issue remains controversial and would take us afield of the primary purpose of the paper, and we therefore leave it to further research. Nonetheless, we recognize that by abstracting from other shocks, our quantitative results regarding welfare costs under alternative policies likely dramatically understate those that would obtain if we included a full specification of all shocks that impact the economy. 4.1 Approximating Household Welfare As is now standard in the literature, we approximate household utility with a second-order Taylor expansion around the deterministic steady state. We denote steady-state values with an asterisk subscript. As shown in Appendix D, the second-order approximation of the period utility function depends on the squared output gap (the log difference between output and its natural rate, that is x t = y t ỹ t ), the squared quasi-difference of the output gap, the cross-product of the output gap and its quasi-difference, and the squared price and wage inflation rates. As shown in the appendix, in the linearized model, the natural rate of output, ỹ t, is a function of leads and lags of the technology shock. After numerous manipulations, the second-order approximation to utility can be written 13

15 as ) ζ(1 σ) 1 1 σ (C t ηc t 1 ) 1 σ ( L Lu,t (C ηc ) 1 σ ( L ) ζ(1 σ) Lu, = 1 ( ) 1 ζ(1 σ) 1 βη 2 x 2 t 2 ζ 1 η 1 2 σ ) 2 (x (1 η) 2 t ηx t 1 1 βη ) (1 σ) (1 η) 2 x t (x t ηx t βη 1 η + T.I.P., { θp Π p, κ p π 2 p,t + θ wπ l, κ w π 2 w,t } where T.I.P refers to terms that are independent of monetary policy. In the following, we ignore these terms in our calculations and focus on the terms related to the output gap and price and wage inflation rates. The first three right-hand side terms correspond to the welfare costs associated with output deviating from its natural rate. Owing to the presence of habit formation, both the level of the output gap and its quasi-difference affect welfare. Note that all three preference parameters enter in the coefficients of the welfare loss for these three terms. The final term corresponds to the welfare costs associated with adjustment costs in changing prices and wages. The coefficients in these terms depend primarily on the parameters associated with nominal rigidities. Importantly, the welfare costs of sticky prices and wages are inversely related to the price and wage sensitivity parameters, κ p and κ w, respectively. The more flexible are prices, the smaller are the welfare costs associated with a given magnitude of inflation fluctuations, and similarly for wages. In the following, we denote the welfare loss abstracting from the terms of independent of policy by L. We also report the three components of the loss, L x, L p, and L w, that correspond to the components of the welfare loss associated with output gaps (and their quasi-differences), price inflation, and wage inflation, respectively. Table 2 reports the implied relative weights on the terms related to the output gap, wage inflation, and price inflation. The first row reports the sum of the weights on the three terms in the loss associated with the output gap and its quasi-difference. 7 For this table, we 7 Based on the point estimates, the weights on the squared level of the output gap and the squared quasi- 14

16 Table 2: Relative Weights in Central Bank Loss Weight Point Mean Standard Correlation with in Loss Estimate Value Deviation ω x ω w ω p ω x ω p ω w have normalized the values of the weights by the weight on price inflation evaluated at the parameter point estimates. The first column reports the weights based on the parameter point estimates. The second column reports the mean values of the weights based on the estimated distribution of the parameter values, approximated using 1 draws from the normal distribution with the estimated covariance for the parameter estimates, where we truncate the parameter values at the lower ends of their distributions as follows: σ at.5, ζ at.1, and η at. The third column reports the corresponding standard deviations of the weights. The final three columns report the cross-correlation of the weights. Based on the point estimates, the variance in wage inflation gets a weight of over 1 times that of price inflation in the welfare loss owing to the estimated value of κ w being one tenth as large as that for κ p. The weights on the variances of the output gap and the quasidifference of the output gap are somewhat smaller than that of inflation, but are somewhat higher than typically seen in the literature owing to our relatively high estimate of σ. The mean values of the weights exhibit the same pattern, but are between two and three times larger than those based on the point estimates, reflecting the fact that the weights depend in part on the inverse of some parameter values. The relative weights are highly positively correlated, indicating that a large weight on output gap terms is associated with large weights on price and wage inflation terms. This high degree of correlation reflects that fact that each component of the welfare loss depends on the steady-state level of utility. difference of the output gap are about equal, while that on the cross-product is smaller and has the opposite sign. 15

17 4.2 Optimal Monetary Policy with No Uncertainty To compute the optimal certainty equivalent policy for a given set of parameter values, we maximize the quadratic approximation of welfare subject to the constraints implied by the linearized model. Throughout, in computing the welfare loss we assume a discount rate arbitrarily close to zero, so that we are maximizing the unconditional measure of welfare. We compute the fully optimal policy using Lagrangian methods as described in Finan and Tetlow (1999). We assume that the technology shock is the only stochastic element in the model and calibrate the standard deviation of its innovations to equal.64 percentage point. This value is slightly larger than the corresponding estimate in LOWW (25). The results under the fully optimal policy are shown in the first column of Table 3. The middle portion of the table shows the welfare loss and the breakdown into its component parts (the components add to the total welfare loss, subject to rounding). 8 Note that we do not normalize the welfare loss in this table or in those that follow. The lower part of the table reports the resulting unconditional standard deviations of output gap, price and wage inflation rates, and the nominal interest rate. The remaining entries in the table are discussed below. Under the fully optimal monetary policy, output gap and wage inflation variability are reduced to nearly zero, while some price inflation variability remains. In terms of the annualized rate, the standard deviation of price inflation is.8 percentage points, about 1 times greater than that of wage inflation. Note that this policy induces considerable interest rate variability in response to a single source of shocks, with the standard deviation of the nominal interest rate 3.6 percentage points on an annualized basis. 9 Under the optimal policy, variability in price inflation accounts for most of the welfare loss. 4.3 Implementable Monetary Policy Rules The fully optimal monetary policy can be implemented in a number of equivalent ways if all model parameters are known with certainty. In the presence of parameter uncertainty, we need to restrict ourselves to representations of monetary policy that are constrained by 8 The welfare losses are in absolute terms. If measured in permanent consumption equivalent units, these losses are very small, reflecting the fact that we are only considering one source of shocks to the economy. 9 In the presence of other shocks, the optimal policy likely produces interest rate variability so great that the zero lower bound on nominal interest rate surely becomes a relevant concern. We leave the analysis incorporating this constraint to future work. 16

18 Table 3: Performance of Alternative Monetary Policies without Uncertainty Optimal Policy Rule Policy Coefficients r n x 1. l 1.93 π p π w Welfare Losses L L x L p L w Standard deviations x π p π w r the information set that the policymaker possesses. For this purpose, we choose to study monetary policies in terms of feedback or instrument rules where the short-term interest rate is determined by a small number of observable variables and the coefficients of the policy rule are chosen to minimize the welfare loss. We consider four different specifications of monetary policy, each of which yields welfare very close to the fully optimal policy when all parameters are known. The general specification is given by a Taylor-type monetary policy rule where the nominal interest rate is determined by the central bank estimate of the natural rate of interest, r t n, the central bank estimate of the output gap, x t, the level of hours, l t, and the rates of price and wage inflation: r t = π p,t + φ r n r t n + φ x x t + φ l l t + φ p π p,t + φ w π w,t. (12) Note that we have included the price inflation rate as the first term of the equation, implying that the policy yields a unique rational expectations equilibrium as long as one of the other 17

19 coefficients (on price inflation, wage inflation, the output gap, or the level of hours) is strictly positive. The response to hours is assumed to be to the level of hours, not the hours gap. With known parameters, the central bank estimates of the natural rates equal their respective true values. We start by considering a textbook Taylor rule with a unit response to the natural rate of interest and a free coefficients on the rates of price inflation and the output gap. We optimized the coefficients of this rule to maximize unconditional welfare of the representative household using a numerical hill-climber routine, as described in Levin, Wieland, and Williams (1999). Throughout the following, we restrict policy rule coefficients to be non-negative and to not exceed an upper bound of 1. 1 The results for the optimized Taylor rule are given in the second column in the table. The optimized Taylor rule has a small coefficient on the price inflation rate and the maximal allowable coefficient on the output gap. This rule strives to keep the output gap at zero. The resulting outcome yields a welfare loss nearly identical to the fully optimal policy, with very slightly too much wage inflation variability. We next consider a variant of the Taylor rule where policy responds to the rate of wage inflation instead of the output gap. This rule features a zero optimal response to price inflation and a moderate response to wage inflation. The resulting outcomes and corresponding welfare are virtually identical to those under the fully optimal policy. Finally, we consider two alternative specifications of the monetary policy rule that do not depend on estimates of the natural rates of output or interest. Interestingly, in both cases, the optimized versions of these rules come very close to mimicing the outcomes under the fully optimal policy. First, we consider a rule that does not respond to the natural rate of interest (except for its long-run mean) or any measure of economic activity, but instead responds only to price and wage inflation. The optimized parameterization of this rule features huge responses to price and wage inflation with the response to wage inflation about 1 times larger than that to price inflation. Second, we consider a rule that responds to price inflation and the log-level of hours. For this rule, the optimized response to price 1 In the two cases where this upper bound is a binding constraint, the loss surface is nearly flat in the vicinity of the reported parameter values. In the case of the Taylor Rule, increasing the upper bound to 1, has no effect on the loss (at three decimal places); in the case of the rule that only responds to wage and price inflation, relaxing the constraint lowers the loss from 1.76 to 1.758, the same as that under the fully optimal policy (at three decimal places). 18

20 inflation is zero and that to hours is about 2. 5 Monetary Policy under Parameter Uncertainty In this section, we analyze the performance and robustness of various monetary policies under parameter uncertainty. We assume that the central bank knows the true model and that the model is estimated using a consistent estimator and that the central bank is certain that the model and the estimation methodology are correct. 11 The only form of uncertainty facing the policymaker is uncertainty regarding model parameters owing to sample variation. We abstract from learning and assume that the policymaker s uncertainty does not change over time. For a given policy rule, expected welfare is computed by numerically integrating over the distribution of the five estimated structural parameters as measured by the estimated covariance matrix. Note that in these calculations, we fully take into account the effects of parameter values on the parameters of the loss function as in Levin and Williams (25). 5.1 Natural Rate Uncertainty Before proceeding with the analysis of policy rules, we first provide some summary measures of the degree of uncertainty regarding the natural rates of hours, output, and interest owing to parameter uncertainty. In this model, the responses of the natural rates to a technology shock depend on three parameters describing household preferences: σ, η, and ζ. Throughout the following, we assume that the distribution of model parameters is jointly normal distributed with the estimated covariance matrix. In the following, we approximate this distribution with a large number of draws from the estimated covariance matrix. Parameter uncertainty implies uncertainty regarding the responses of natural rates to technology shocks. The thin solid line in the upper panel of Figure 3 plots the impulse response of the log of the natural rate of hours to a one percentage point positive permanent technology shock based on the point estimates of the model parameters. (Note that the log 11 The assumption that the policymaker is certain about the correctness of the estimation methodology likely reduces the degree of parameter uncertainty relative to what policymakers face in reality. For example, in the model used in this paper, some parameter point estimates can vary significantly, depending on sample and specifics of the estimation method. We leave the study of this broader form of estimation uncertainty to future work. 19

21 of the natural rate of hours equals the log of the natural rate of output minus the log of TFP.) The thick solid line shows the median response calculated from impulse responses corresponding to 1, draws from the estimated parameter distribution. The dashed and dashed-dotted lines show the boundaries of the 7 and 9 percent confidence bands of the impulse responses, respectively. The lower panel of the figure shows the corresponding outcomes for the natural rate of interest measured at an annualized rate. Note that the model implies that there is no uncertainty about the long-run effects of technology shocks on the natural rates of hours and interest, both of which eventually return to their respective steady state values. Natural rate misperceptions owing to parameter uncertainty are sizable and persistent. We measure natural rate misperceptions as the difference between the level of the natural rate implied by the model s true parameter values and the level implied by the point estimates of the model parameters. 12 To compute unconditional moments in our model, we calibrate the standard deviation of innovations to technology to equal.64 percentage point, equal to the sample average of the identified shocks from our VAR. The resulting unconditional standard deviation of the difference between the true natural rate of output and the central bank s estimate (based on the model with the parameter point estimates) is.32 percentage point. The first-order autocorrelation of this difference is.86. The unconditional standard deviation of the difference between the natural rate of interest and the central bank s estimate is 3.15 percentage points (measured at an annual rate), with a first-order autocorrelation of this difference equal to Optimal Monetary Policy under Parameter Uncertainty In order to provide a benchmark for policies under uncertainty, we first compute the optimal outcome if the policymaker knew all the parameter values and followed the fully optimal policy in each case. We average the outcomes and losses over 1 draws of the parameters and report the results in the first column of Table 4. Of course, given that the parameters are uncertain, this outcome is not obtainable in practice, but provides a benchmark against which we can measure the costs associated with parameter uncertainty. As can be seen from 12 In principle, the central bank could use other methods to estimate the natural rates that take into account parameter uncertainty, but basing the central bank estimates on those implied by the parameter point estimates seems a reasonable benchmark for our analysis. 2

22 Table 4: Performance of Alternative Monetary Policies with Parameter Uncertainty Optimal Policy Rule Policy Coefficients r n x l π p π w Welfare Losses L L x L p L w Standard deviations x π p π w r comparing the first columns of Tables 3 and 4, the mean welfare loss under the first-best optimal policy is considerably larger than that computed at the parameter point estimates. This reflects the fact that the mean weights in the welfare loss are higher than the weights evaluated at the point estimates. Indeed, under the first-best optimal policy, the variability of the objective variables is about the same as for the case of the parameter point estimates. We now examine the characteristics and performance of the implementable monetary policy rules introduced in the previous section, but now we reoptimize the coefficients to minimize the expected welfare loss under parameter uncertainty. In implementing these rules, we assume that the central bank s estimates of the natural rates of output and interest are computed using the point estimates of the model parameters, but that the actual model parameters and therefore natural rates differ from the values assumed by the policymaker. The central bank is assumed to observe the technology shocks without error since these do not depend on model parameters. Relative to the case of no parameter uncertainty, the optimized standard Taylor rule 21